Problem 1 - Algebra 2

Question

Use a double-angle identity to find the exact value of each expression. $$ \sin 240^{\circ} $$

Step-by-Step Solution

Answer

\(\sin(240^\circ) = -\frac{\sqrt{3}}{2}\).

Step 1: Identify a Known Angle and its Relation to the Given Angle

We recognize that 240° is 2 times 120°. We can use the double-angle identity for sine which is \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\) where \(\theta = 120°\).

Step 2: Apply the Sine Double-Angle Identity

Using the double-angle identity, we have \(\sin(240°) = \sin(2\times120°) = 2\sin(120°)\cos(120°)\).

Step 3: Evaluate \(\sin(120°)\) and \(\cos(120°)\)

\(\sin(120°) = \sin(180° - 60°) = \sin(60°)\) which is \(\frac{\sqrt{3}}{2}\). \(\cos(120°) = \cos(180° - 60°) = -\cos(60°)\) which is \( -\frac{1}{2}\).

Step 4: Compute the Exact Value

Now multiply the values together: \( 2\sin(120°)\cos(120°) = 2\times\frac{\sqrt{3}}{2}\times(-\frac{1}{2}) = -\frac{\sqrt{3}}{2}\).

Step 5: Provide the Final Answer

Therefore, the exact value of \(\sin(240°)\) is \( -\frac{\sqrt{3}}{2} \).

Key Concepts

Exact Value of Trigonometric Functions

Understanding the exact value of trigonometric functions is fundamental in mathematics, specifically when dealing with certain angles that have well-known trigonometric values. An angle, usually measured in degrees or radians, plays a crucial role in the evaluation of sine, cosine, and tangent functions. In mathematics, the exact values of trigonometric functions for certain standard angles, such as 0°, 30°, 45°, 60°, and 90°, can be determined using the unit circle or trigonometric ratios that stem from special right triangles, such as the 30-60-90 and 45-45-90 triangles.

When the angle in question is not a standard angle, we can often manipulate it to express it in terms of the standard angles. For instance, the angle used in the exercise, 240°, is not a standard angle, but it can be associated with the standard angle 120°. Recognizing these connections allows us to leverage the known values from the unit circle to find exact trigonometric values for a wider range of angles.

Sine Double-Angle Identity

The sine double-angle identity is a powerful tool in trigonometry that relates the sine of a double angle to the product of the sine and cosine of the original angle. The identity is given by the equation \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\). This identity enables us to break down complex expressions into simpler terms we can manage more easily, particularly when evaluating the sine function at unconventional angles.

For example, if we are asked to find the exact value of \(\sin(240^\circ)\), we can utilize the identity by recognizing that 240° is twice 120°. Applying the double-angle identity simplifies the task from finding \(\sin(240^\circ)\) directly to calculating \(2\sin(120^\circ)\cos(120^\circ)\), which can then be evaluated using known exact values. Moreover, understanding this identity is essential when simplifying trigonometric expressions or solving trigonometric equations.

Evaluating Trigonometric Expressions

Evaluating trigonometric expressions involves substituting given angles into trigonometric functions and using identities or known values to simplify or find the result. A methodical approach to this process usually involves identifying any special angles, leveraging identities like the double-angle formulas, and applying exact values from the unit circle or Pythagorean identities.

As applied in our exercise, after using the double-angle identity to express \(\sin(240^\circ)\) in terms of the sine and cosine of 120°, the next step is to evaluate \(\sin(120^\circ)\) and \(\cos(120^\circ)\), which in this case, are found using their supplementary angles. The process concludes with performing the arithmetic operations to combine these values, producing the exact value for the original expression. It's important to note that signs of the trigonometric values must be carefully managed depending on the quadrant in which the angle lies.